According to Louie Helm eradicating a species of mosquitoes could be done for as little as a few million dollar.
I don't have a few million dollar lying around so I can't spend my own money to do it. On the other hand, I think that on average every German citizen would be quite willing to pay 1€ per year to rid Germany of mosquitoes that bite humans.
That means it's a problem of public action. The German government should spend 80 million Euro to rid Germany of Mosquitos. That's an order of magnitude higher than the numbers quoted by Louie Helm).
The same goes basically for every country or state with mosquitos.
How could we get a government to do this without spending too much money ourselves? The straight forward way is writing a petition. We could host a website and simultaneously post a petition to every relevant parliament on earth.
How do we get attention for the petition? Facebook. People don't like Mosquitos and should be willing to file an internet petition to get rid of them. I would believe this to spread virally. The idea seems interesting enough to get journalists to write articles about it.
Bonus points:
After we have eradicated human biting mosquitoes from our homelands it's quite straightforward to export the technology to Africa.
Does anyone see any issues with that plan?
One issue is the same intervention doesn't necessarily affect both. For example, where I live West Nile virus is transmitted primarily by Culex pippiens mosquitoes, while the most abundant nuisance mosquito is Ochlerotatus stimulans.
Controlling one species will not greatly affect the other (they breed in radically different conditions). It's not a matter of scaling up operations; you need an entirely different strategy, with commensurate increase in operating costs, complexity, potential failure points, etc etc.
Give me unlimited resources and global remit and I'll take them all out, absent this prioritisation becomes necessary.